C(X) vs. C(X) modulo its socle

F. Azarpanah, O. A. S. Karamzadeh, and S. Rahmati. "C(X) vs. C(X) modulo its socle." Colloquium Mathematicae 111.2 (2008): 315-336. <http://eudml.org/doc/286449>.

@article{F2008,
abstract = {Let $C_\{F\}(X)$ be the socle of C(X). It is shown that each prime ideal in $C(X)/C_\{F\}(X)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim (C(X)/C_\{F\}(X)) ≥ dim C(X)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential ideal E in C(X), we observe that $E/C_\{F\}(X)$ is essential in $C(X)/C_\{F\}(X)$ if and only if the set of isolated points of X is finite. Finally, we characterize topological spaces X for which the Jacobson radical of $C(X)/C_\{F\}(X)$ is zero, and as a consequence we observe that the cardinality of a discrete space X is nonmeasurable if and only if υX, the realcompactification of X, is first countable.},
author = {F. Azarpanah, O. A. S. Karamzadeh, S. Rahmati},
journal = {Colloquium Mathematicae},
keywords = {Goldie dimension; Suslin number; socle; essential ideal; Ue-ring; z-ideal; isolated point},
language = {eng},
number = {2},
pages = {315-336},
title = {C(X) vs. C(X) modulo its socle},
url = {http://eudml.org/doc/286449},
volume = {111},
year = {2008},
}

TY - JOUR
AU - F. Azarpanah
AU - O. A. S. Karamzadeh
AU - S. Rahmati
TI - C(X) vs. C(X) modulo its socle
JO - Colloquium Mathematicae
PY - 2008
VL - 111
IS - 2
SP - 315
EP - 336
AB - Let $C_{F}(X)$ be the socle of C(X). It is shown that each prime ideal in $C(X)/C_{F}(X)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim (C(X)/C_{F}(X)) ≥ dim C(X)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential ideal E in C(X), we observe that $E/C_{F}(X)$ is essential in $C(X)/C_{F}(X)$ if and only if the set of isolated points of X is finite. Finally, we characterize topological spaces X for which the Jacobson radical of $C(X)/C_{F}(X)$ is zero, and as a consequence we observe that the cardinality of a discrete space X is nonmeasurable if and only if υX, the realcompactification of X, is first countable.
LA - eng
KW - Goldie dimension; Suslin number; socle; essential ideal; Ue-ring; z-ideal; isolated point
UR - http://eudml.org/doc/286449
ER -